In this paper, we consider Cartan–Hadamard manifolds (i.e., simply connected, complete, of non-positive sectional curvature) whose negative Ricci curvature grows polynomially at infinity. We show that a number of functional properties, which typically hold on manifolds of bounded curvature, remain true in this setting. These include the characterization of Sobolev spaces on manifolds, the so-called Calderón–Zygmund inequalities and the Lp-positivity preserving property, i.e., u∈Lp&(-Δ+1)u≥0⇒u≥0. The main tool is a new class of first- and second-order Hardy-type inequalities on Cartan–Hadamard manifolds with a polynomial upper bound on the curvature. In the last part of the manuscript we prove the Lp-positivity preserving property, p∈[1,+∞], on manifolds with subquadratic negative part of the Ricci curvature. This generalizes an idea of B. Güneysu and gives a new proof of a well-known condition for the stochastic completeness due to P. Hsu.

Marini, L., Veronelli, G. (2024). Some Functional Properties on Cartan–Hadamard Manifolds of Very Negative Curvature. THE JOURNAL OF GEOMETRIC ANALYSIS, 34(4) [10.1007/s12220-023-01541-1].

Some Functional Properties on Cartan–Hadamard Manifolds of Very Negative Curvature

Marini L.
;
Veronelli G.
2024

Abstract

In this paper, we consider Cartan–Hadamard manifolds (i.e., simply connected, complete, of non-positive sectional curvature) whose negative Ricci curvature grows polynomially at infinity. We show that a number of functional properties, which typically hold on manifolds of bounded curvature, remain true in this setting. These include the characterization of Sobolev spaces on manifolds, the so-called Calderón–Zygmund inequalities and the Lp-positivity preserving property, i.e., u∈Lp&(-Δ+1)u≥0⇒u≥0. The main tool is a new class of first- and second-order Hardy-type inequalities on Cartan–Hadamard manifolds with a polynomial upper bound on the curvature. In the last part of the manuscript we prove the Lp-positivity preserving property, p∈[1,+∞], on manifolds with subquadratic negative part of the Ricci curvature. This generalizes an idea of B. Güneysu and gives a new proof of a well-known condition for the stochastic completeness due to P. Hsu.
Articolo in rivista - Articolo scientifico
35A23; 35J10; 46E35; 58J65; Calderón–Zygmund inequalities; Cartan–Hadamard manifolds; Hardy-type inequalities; L; p; -positivity preserving property; Primary: 53C21; Secondary: 58J05; Sobolev spaces on manifolds; Stochastic completeness;
English
20-feb-2024
2024
34
4
106
partially_open
Marini, L., Veronelli, G. (2024). Some Functional Properties on Cartan–Hadamard Manifolds of Very Negative Curvature. THE JOURNAL OF GEOMETRIC ANALYSIS, 34(4) [10.1007/s12220-023-01541-1].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/476120
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